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International Centre for Theoretical Physics :L^ -^ . ::, i^C IC/89/126 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Eric Bergshoeff and Ergin Sezgin INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION 1989 MIRAMARE - TRIESTE IC/89/I2G It is well known that supcrslrings admit two different kinds of formulations as far as International Atomic Energy Agency llieir siipcrsytnmclry properties arc concerned. One of them is the Ncveu-Sdwarz-tUmond and (NSIt) formulation [I] in which the world-sheet supcrsymmclry is manifest but spacetimn United Nations Educational Scientific and Cultural Organization supcrsymmclry is not. In this formulation, spacctimc supcrsymmctry arises as a consequence of Glbzzi-Schcrk-Olive (GSO) projections (2J in the Hilbcrt space. The field equations of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS the world-sheet graviton and gravilino arc the constraints which obey the super-Virasoro algebra. In the other formulation, due to Green and Schwarz (GS) [3|, the spacrtimc super- symmetry is manifest but the world-sheet supersymmclry is not. In this case, the theory lias an interesting local world-sheet symmetry, known as K-symmetry, first discovered by NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Sicgcl [4,5) for the superparticlc, and later generalized by Green and Schwarz [3] for super- strings. In a light-cone gauge, a combination of the residual ic-symmetry and rigid spacetime supcrsyininetry fuse together to give rise to an (N=8 or 1G) rigid world-sheet supersymmclry. Eric BergflliocIT The two formulations are essentially equivalent in the light-cone gauge [6,7]. How- Theory Division, CERN, 1211 Geneva 23, Switzerland ever, for many purposes it is desirable to have a covariant quantization scheme. The covariant quantization of the NSR spinning string is well understood. In many modern treatments of and the theory, an increasingly central role is played by the NSR super-Virasoro algebra. In the case of GS superstring, one of the important questions which naturally arises is to find the Ergin Sezgin analog of the NSR super- Virasoro algebra. International Center for Theoretical Physics, Trieste, Italy Sicgcl has argued that [8,9] a generalized super-Virasoro algebra for the GS string can be arrived at in two steps: 1. Find a set of operators {"supercovariant derivatives") satisfying a generalized Kac-Moody algebra. 2. Find a suitable set of constraints quadratic in these ABSTRACT operators satisfying a generalized Virasoro algebra ("Sugawara construction"). Sicgel \8,9] Achieved this construction for super particles and supcrstrings. Two salient features of his results arc: (a) All constraints arc first class. In particular, among the constraints is the We present new spacclime algebras whose existence is due to special f-matrix iden- one corresponding to the K-symmetry. (b) The supercovariant derivative corresponding to tities which are also necessary lor the existence of super p-brancs. They contain a p-lh rank superstranslations does not commute with the bosonic translations. antisymmetric tensor, and a (p-l) rank antisymmetric tensor-spinor generator. Furthermore The feature (b) above is rather interesting: Recently, Green [10] pointed out that the the translations do not commute with the supercharge. In the case ofsupennrmbranes, wo global limit of the Sicgcl algebra is a new (differing from the super-Poincarc) supcralgcbra in find a su per-Kac-Moody extension of the new spaccttmc algebra. We give a realisation of whicli llic bosonic translations do not commute witlt supcrsymmctry. This new supcralgcbra the algebra in terms of operators in d=l 1 supenpace, and find a connection with the d=I 1 can be naturally described by an extended superspace which contains new fermionic coor- supcrmembrane action which leads to an elegant supcrgeometric formulation. Hy double dinates. An attractive feature of this supcralgcbra is that its associated super-Killing form dimensional reduction, we obtain a Kac-Moody algebra For the Type IIA Grcen-Schwarz is nondegenerate. This fact has been exploited by Green (10) to find a (2+l)-dimensional supcrstring. We also discuss the generalization of these results, including the construction Chern-Simons formulation of the GS supcrstring. of generalized super- Virasoro algebras, for super p-brancs. In lilts note, generalizing Green's result, we find new spacctimc supnralgrbras in all dimensions where super p-bram-s exist [l I-1:J], i.e. in addition to tlie superstring case (p=l; MIRAMARE.THIRSTK <l=:j,'l,fi,10), we find that new spiurtiim' superalgebras also exist for the supermembrancs July 1989 (|)=2; 41=4,5,7,11), and the other super p-bratw-s: (p=.1; d=6,8), (p=4; d=9) and (p=5; T d=10). In particular, we shall see that (a) the new superalgebras contain a p-th rank anti- where 6V 8 = SP{V"m)a0f. These operators are invariant under translations and super- symmetric tensor, and & (p-l)-rank antisymmetric tensor-spinor generator, (b) the bosonic symmetry which are generated by [8,9] translations do not commute with supersytnmetry, and (c) the existence of the algebras rely on r-matrix identities which are precisely the ones needed for the existence of the super p-branes. (3) In this paper, we shall focus our attention on the most interesting case of the eleven dimensional supermembrane, and construct for that case a generalized Kac-Moody super- These generators obey the usual super-Poincare algebra. However, Green [10] has recently algebra. We shall also find the realization of the algebra in terms of operators in d=l 1 observed that, one can write down a new spacetime superalgebra with the following nonva- superspace. Here, a surprising connection with the d=ll supermembrane action emerges in nishing commutators and anticommutators that the same super 3-form which plays a central role in the construction of the K-invari&nt supermembrane action [12] also arises in the superspace realization of the Kac-Moody gen- "* e (4) erators. In fact, this leads to a remarkably simple and elegant supergeometric formulation of the full Kac-Moody algebra. where A'o is a new fermionic generator. This algebra exists only in d=3,4,6 and 10. Note We shall also suggest a way to construct the constraints (including those of rcparamelriza- that we can include the Lorentz generators, MM,, as well. We would then add the usual tions and K-symmetry ) which are quadratic in the Kac-Moody generators, but we shall not commutators: [M,Mj ~ M; [M,P] ~ P\ \M,Q\ ~ Q; \M,U\ ~ tt and \P,P\ = 0. compute their full algebra. In the next section we deacribe briefly the results of Siegel [8] and The algebra (4) can be obtained as the global limit of (1). The global generators Green [10J for the GS superstring, pertaining to the new spacetime algebra, its Kac-Moody P,Q and K can be realized as follows [10]: extension and the generalized super-Virasoro algebra. After that we present our results for supermembranes, and Type IIA superstrings. Generalizations to other super p-branes are also discussed. The Green-Schwarz Superstring (5) To describe the classical mechanics of the GS superstring, Siegel introduces the operators Ppt and Da, corresponding to bosonic and fermionic translations in d=3,4,6 or 10 superspace respectively, and a new operator ft*, which obeys the following Kac-Moody where -V, 0° and <j>" are the coordinates of an extended superspace. The algebra (4) has super&lgebra [8] a nonvanishing Killing form. This fact has been used by Green [10] to construct a d=3 Chern-Simons gauge theory formulation of the GS superstring. In the quantization of the theory on certain Riemann surfaces, one may have to make the identification jf — d\$, in (1) which case the algebra of the generators (5) is the usual super-Poincare algebra [10]. To construct a generalized Virasoro algebra for the GS superstring, Siegel introduces with other commutators and anticommutators vanishing, and d\ = djda. the following generators [8] One can check that the Jacob! identities are satisfied by (1), provided that the V- matrix identity r^^r'w = 0 holds. This is possible only in d=3,4,6 and 10. The algebra (1) can be realized in terms of the following operators [8,9]: CBff = DaDg, The generators A, B, C and V obey the sought generalized Virasoro algebra [8], In particular, (2) the generator A obeys the usual Virasoro algebra: The remaining part of the algebra is very complicated. It has been studied in detail also by with all the other commutators and anticommutators vanishing. The Lorentz generators, Romans [14], Schematically, it is given by Af,,, can be included, with the obvious commuUton: [M,M] ~ M\[M,P] ~ P;[M,A] ~ A;[M,Q] ~ O;(Af,nj ~ n and \P,P\ = 0. The dimensions and the reality properties of ~ f'B, \AC\ ~ 8C, the spinor Q" is spelled out in detail case by case in |16]. The properties of K~ then {B, 8) ~ #C + 6{PA + V + 08), \BA = 6(DA + OC), follow trivially. One can easily check that the Jacobi identities are indeed satisfied, and {B, D) = g(DA + OC) + f(&A + UC+ nV), (8) in particular the {Q, {Q,Q}} Jacobi identity does require the F-matrix identities (9). We P,C] = 6PC, [C, 1>] = SPC + &{&B + PC'), expect the algebra (10) to have a number of interesting consequences which we hope to report {V, V] = fTDB + tf(PV + 0B + fC) + S(P'V + &B1 + PC), on in the future. In the rest of this paper, we shall present our results on the generalization of (1), (2) and (3) for the case of most interest, namely the eleven dimensional supermembrane, where £ = S(tr - </) and ^ = &iS{o — <r*).
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